{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T15:42:21Z","timestamp":1758814941176},"reference-count":18,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,5]],"date-time":"2006-10-05T00:00:00Z","timestamp":1160006400000},"content-version":"vor","delay-in-days":6243,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1989,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Three obvious necessary conditions for the existence of a <jats:italic>k<\/jats:italic>\u2010cycle system of order <jats:italic>n<\/jats:italic> are that if <jats:italic>n<\/jats:italic> &gt; 1 then <jats:italic>n<\/jats:italic> \u2a7e <jats:italic>k, n<\/jats:italic> is odd, and 2<jats:italic>k<\/jats:italic> divides <jats:italic>n<\/jats:italic>(<jats:italic>n<\/jats:italic> \u2212 1). We show that if these necessary conditions are sufficient for all <jats:italic>n<\/jats:italic> satisfying <jats:italic>k<\/jats:italic> \u2a7d <jats:italic>n<\/jats:italic> &lt; 3<jats:italic>k<\/jats:italic> then they are sufficient for all <jats:italic>n.<\/jats:italic> In particular, there exists a 15\u2010cycle system of order <jats:italic>n<\/jats:italic> if and only if <jats:italic>n<\/jats:italic> \u2261 1, 15, 21, or 25 (mod 30), and there exists a 21\u2010cycle system of order <jats:italic>n<\/jats:italic> if and only if <jats:italic>n<\/jats:italic> \u2261 1, 7, 15, or 21 (mod 42), <jats:italic>n<\/jats:italic> \u2260 7. 15.<\/jats:p>","DOI":"10.1002\/jgt.3190130405","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T12:25:32Z","timestamp":1180182332000},"page":"417-426","source":"Crossref","is-referenced-by-count":49,"title":["On the construction of odd cycle systems"],"prefix":"10.1002","volume":"13","author":[{"given":"D. G.","family":"Hoffman","sequence":"first","affiliation":[]},{"given":"C. C.","family":"Lindner","sequence":"additional","affiliation":[]},{"given":"C. A.","family":"Rodger","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,5]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-5060(08)70053-0"},{"key":"e_1_2_1_3_2","first-page":"53","article-title":"Graph decompositions and G\u2010designs","volume":"15","author":"Bermond J. C.","year":"1975","journal-title":"Congressus Numerantium"},{"key":"e_1_2_1_4_2","first-page":"293","article-title":"Balanced cycle and circuit designs: Even case","volume":"5","author":"Bermond J. C.","year":"1978","journal-title":"Ars. Combinat."},{"key":"e_1_2_1_5_2","unstructured":"J. C.BermondandD.Sotteau Cycle and circuit designs: Odd Case. Beitrage zur Graphentheorie und deren Anwendungen. Proc. Colloq. Oberhof Illmenau (1978)11\u201332."},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-349-03521-2"},{"key":"e_1_2_1_7_2","unstructured":"C. K.Fu The intersection problem for pentagon systems. Ph.D. thesis Auburn University (1987)."},{"key":"e_1_2_1_8_2","first-page":"227","article-title":"A lemma on cycle decompositions","volume":"27","author":"H\u00e4ggkvist R.","year":"1985","journal-title":"Ann. Discrete Math."},{"key":"e_1_2_1_9_2","first-page":"227","article-title":"On decompositions of complete graphs into 4k\u2010gons","volume":"15","author":"Kotzig A.","year":"1965","journal-title":"Mat.\u2010 Fyz. Cas"},{"key":"e_1_2_1_10_2","first-page":"349","article-title":"On cyclic decompositions of the complete graph into (4m + 2)\u2010gons","volume":"16","author":"Rosa A.","year":"1966","journal-title":"Mat.\u2010Fyz. Cas"},{"key":"e_1_2_1_11_2","first-page":"53","article-title":"On the cyclic decomposition of the complete graph into polygons with an odd number of edges","volume":"91","author":"Rosa A.","year":"1966","journal-title":"Casopis Pest. Math."},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(75)90051-5"},{"key":"e_1_2_1_13_2","unstructured":"D.Sotteau D\u00e9compositions de graphes et hypergraphes. Th\u00e8ses de l'Universit\u00e9 Paris\u2010Sud (1980)."},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(81)90093-9"},{"key":"e_1_2_1_15_2","first-page":"109","article-title":"Steiner triple systems with given subspaces: Another proof of the Doyen\u2010Wilson theorem","volume":"17","author":"Stern G.","year":"1980","journal-title":"Boll. Un. Math. Ital. A(5)"},{"key":"e_1_2_1_16_2","first-page":"25","article-title":"On an estimate of the chromatic class of a p\u2010graph","volume":"3","author":"Vizing V. G.","year":"1964","journal-title":"Discrete Analiz."},{"key":"e_1_2_1_17_2","first-page":"18","article-title":"Construction and uses of pairwise balanced designs","volume":"55","author":"Wilson R. M.","year":"1974","journal-title":"Math. Centre Tracts"},{"key":"e_1_2_1_18_2","first-page":"647","article-title":"Decompositions of complete graphs into subgraphs isomorphic to a given graph","volume":"15","author":"Wilson R. 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